不等式 英文版PDF电子书下载

1 Measure and integral 4

1.1 Measure 4

1.2 Measurable functions 7

1.3 Integration 9

1.4 Notes and remarks 12

2 The Cauchy-Schwarz inequality 13

2.1 Cauchy's inequality 13

2.2 Inner-product spaces 14

2.3 The Cauchy-Schwarz inequality 15

2.4 Notes and remarks 17

3 The AM-GM inequality 19

3.1 The AM GM inequality 19

3.2 Applications 21

3.3 Notes and remarks 23

4 Convexity,and Jensen's inequality 24

4.1 Convex sets and convex functions 24

4.2 Convex functions on an interval 26

4.3 Directional derivatives and sublinear functionals 29

4.4 The Hahn-Banach theorem 31

4.5 Normed spaces,Banach spaces and Hilbert space 34

4.6 The Hahn-Banach theorem for normed spaces 36

4.7 Barycentres and weak integrals 39

4.8 Notes and remarks 40

5 The Lp spaces 45

5.1 Lp spaces,and Minkowski's inequality 45

5.2 The Lebesgue decomposition theorem 47

5.3 The reverse Minkowski inequality 49

5.4 H？lder's inequality 50

5.5 The inequalities of Liapounov and Littlewood 54

5.6 Duality 55

5.7 The Loomis-Whitney inequality 57

5.8 A Sobolev inequality 60

5.9 Schur's theorem and Schur's test 62

5.10 Hilbert's absolute inequality 65

5.11 Notes and remarks 67

6 Banach function spaces 70

6.1 Banach function spaces 70

6.2 Function space duality 72

6.3 Orlicz spaces 73

6.4 Notes and remarks 76

7 Rearrangements 78

7.1 Decreasing rearrangements 78

7.2 Rearrangement-invariant Banach function spaces 80

7.3 Muirhead's maximal function 81

7.4 Majorization 84

7.5 Calderón's interpolation theorem and its converse 88

7.6 Symmetric Banach sequence spaces 91

7.7 The method of transference 93

7.8 Finite doubly stochastic matrices 97

7.9 Schur convexity 98

7.10 Notes and remarks 100

8 Maximal inequalities 103

8.1 The Hardy-Riesz inequality（1＜p＜∞） 103

8.2 The Hardy-Riesz inequality(p=1) 105

8.3 Related inequalities 106

8.4 Strong type and weak type 108

8.5 Riesz weak type 111

8.6 Hardy,Littlewood,and a batsman's averages 112

8.7 Riesz's sunrise lemma 114

8.8 Differentiation almost everywhere 117

8.9 Maximal operators in higher dimensions 118

8.10 The Lebesgue density theorem 121

8.11 Convolution kernels 121

8.12 Hedberg's inequality 125

8.13 Martingales 127

8.14 Doob's inequality 130

8.15 The martingale convergence theorem 130

8.16 Notes and remarks 133

9 Complex interpolation 135

9.1 Hadamard's three lines inequality 135

9.2 Compatible couples and intermediate spaces 136

9.3 The Riesz—Thorin interpolation theorem 138

9.4 Young's inequality 140

9.5 The Hausdorff—Young inequality 141

9.6 Fourier type 143

9.7 The generalized Clarkson inequalities 145

9.8 Uniform convexity 147

9.9 Notes and remarks 150

10 Real interpolation 154

10.1 The Marcinkiewicz interpolation theorem：Ⅰ 154

10.2 Lorentz spaces 156

10.3 Hardy's inequality 158

10.4 The scale of Lorentz spaces 159

10.5 The Marcinkiewicz interpolation theorem：Ⅱ 162

10.6 Notes and remarks 165

11 The Hilbert transform,and Hilbert's inequalities 167

11.1 The conjugate Poisson kernel 167

11.2 The Hilbert transform on L2(R) 168

11.3 The Hilbert transform on Lp(R)for 1＜p＜∞ 170

11.4 Hilbert's inequality for sequences 174

11.5 The Hilbert transform on T 175

11.6 Multipliers 179

11.7 Singular integral operators 180

11.8 Singular integral operators on Lp(Rd)for 1≤p＜∞ 183

11.9 Notes and remarks 185

12 Khintchine's inequality 187

12.1 The contraction principle 187

12.2 The reflection principle,and Lévy's inequalities 189

12.3 Khintchine's inequality 192

12.4 The law of the iterated logarithm 194

12.5 Strongly embedded subspaces 196

12.6 Stable random variables 198

12.7 Sub-Gaussian random variables 199

12.8 Kahane's theorem and Kahane's inequality 201

12.9 Notes and remarks 204

13 Hypercontractive and logarithmic Sobolev inequalities 206

13.1 Bonami's inequality 206

13.2 Kahane's inequality revisited 210

13.3 The theorem of Latala and Oleszkiewicz 211

13.4 The logarithmic Sobolev inequality on Dd 2 213

13.5 Gaussian measure and the Hermite polynomials 216

13.6 The central limit theorem 219

13.7 The Gaussian hypercontractive inequality 221

13.8 Correlated Gaussian random variables 223

13.9 The Gaussian logarithmic Sobolev inequality 225

13.10 The logarithmic Sobolev inequality in higher dimensions 227

13.11 Beckner's inequality 229

13.12 The Babenko-Beckner inequality 230

13.13 Notes and remarks 232

14 Hadamard's inequality 233

14.1 Hadamard's inequality 233

14.2 Hadamard numbers 234

14.3 Error-correcting codes 237

14.4 Note and remark 238

15 Hilbert space operator inequalities 239

15.1 Jordan normal form 239

15.2 Riesz operators 240

15.3 Related operators 241

15.4 Compact operators 242

15.5 Positive compact operators 243

15.6 Compact operators between Hilbert spaces 245

15.7 Singular numbers,and the Rayleigh-Ritz minimax formula 246

15.8 Weyl's inequality and Horn's inequality 247

15.9 Ky Fan's inequality 250

15.10 Operator ideals 251

15.11 The Hilbert-Schmidt class 253

15.12 The trace class 256

15.13 Lidskii's trace formula 257

15.14 Operator ideal duality 260

15.15 Notes and remarks 261

16 Summing operators 263

16.1 Unconditional convergence 263

16.2 Absolutely summing operators 265

16.3 (p,q)-summing operators 266

16.4 Examples of p-summing operators 269

16.5 (p,2)-summing operators between Hilbert spaces 271

16.6 Positive operators on L1 273

16.7 Mercer's theorem 274

16.8 p-summing operators between Hilbert spaces(1≤p≤2) 276

16.9 Pietsch's domination theorem 277

16.10 Pietsch's factorization theorem 279

16.11 p-summing operators between Hilbert spaces（2≤p ≤∞） 281

16.12 The Dvoretzky-Rogers theorem 282

16.13 Operators that factor through a Hilbert space 284

16.14 Notes and remarks 287

17 Approximation numbers and eigenvalues 289

17.1 The approximation,Gelfand and Weyl numbers 289

17.2 Subadditive and submultiplicative properties 291

17.3 Pietsch's inequality 294

17.4 Eigenvalues of p-summing and(p,2)-summing endomorphisms 296

17.5 Notes and remarks 301

18 Grothendieck's inequality,type and cotype 302

18.1 Littlewood's 4/3 inequality 302

18.2 Grothendieck's inequality 304

18.3 Grothendieck's theorem 306

18.4 Another proof,using Paley's inequality 307

18.5 The little Grothendieck theorem 310

18.6 Type and cotype 312

18.7 Gaussian type and cotype 314

18.8 Type and cotype of Lp spaces 316

18.9 The little Grothendieck theorem revisited 318

18.10 More on cotype 320

18.11 Notes and remarks 323